Expected Value, the Law of Averages, and
the Central Limit TheoremMath 1680
Overview
♠ Chance Processes and Box Models♠ Expected Value♠ Standard Error♠ The Law of Averages♠ The Central Limit Theorem♠ Roulette♠ Craps♠ Summary
Chance Processes and Box Models
♠ Recall that we can use a box model to describe chance processes♥ Flipping a coin♥ Rolling a die♥ Playing a game of roulette
♠ The box model representing the roll of a single die is
1 2 3 4 5 6
Chance Processes and Box Models
♠ If we are interested in counting the number of even values instead, we label the tickets differently
♥ We get a “1” if a 2, 4, or 6 is thrown ♥ We get a “0” otherwise♥ To find the probability of drawing a ticket type
from the box♣ Count the number of tickets of that type♣ Divide by the total number of tickets in the box
♠ We can say that the sum of n values drawn from the box is the total number of evens thrown in n rolls of the dice
0 3 1 3
Expected Value
♠ Consider rolling a fair die, modeled by drawing from
♥ The smallest possible value is 1♥ The largest possible value is 6
1 2 3 4 5 6
Expected Value
♠ The expected value (EV) on a single draw can be thought of as a weighted average ♥ Multiply each possible value by the probability
that value occurs♥ Add these products together
EV1 = (1/6)(1)+(1/6)(2)+(1/6)(3)+(1/6)(4)+(1/6)(5)+(1/6)(6) = 3.5♣ Expected values may not be feasible outcomes
♥ The expected value for a single draw is also the average of the values in the box
Expected Value
♠ If we play n times, then the expected value for the sum of the outcomes is the expected value for a single outcome multiplied by n♥ EVn = n(EV1)
♠ For 10 rolls of the die, the expected sum is 10(3.5) = 35
Expected Value
♠ Flip a fair coin and count the number of heads♥ What box models this game?
♥ How many heads do you expect to get in…
♣10 flips?♣100 flips?
0 1
5
50
Expected Value♠ Pay $1 to roll a fair die
♥ You win $5 if you roll an ace (1)♥ You lose the $1 otherwise
♠ What box models this game?
♠ How much money do you expect to make in…♥ 1 game?♥ 5 games?
♠ This is an example of a fair game
-$1 5 $5 1
$0
$0
Standard Error
♠ Bear in mind that expected value is only a prediction ♥ Analogous to regression predictions
♠ EV is paired with standard error (SE) to give a sense of how far off we may still be from the expected value♥ Analogous to the RMS error for
regression predictions
Standard Error
♠ Consider rolling a fair die, modeled by drawing from
♠ The smallest possible value is 1♥ The largest possible value is 6♥ The expected value (EV) on a single draw is 3.5
♠ The SE for the single play is the standard deviation of the values in the box
1 2 3 4 5 6
71.16
)5.36()5.35()5.34()5.33()5.32()5.31( 222222
1
SE
Standard Error
♠ If we play n times, then the standard error for the sum of the outcomes is the standard error for a single outcome multiplied by the square root of n♥ SEn = (SE1)sqrt(n)
♠ For 10 rolls of the die, the standard error is (1.71)sqrt(10) 5.41
Standard Error
♠ In games with only two outcomes (win or lose) there is a shorter way to calculate the SD of the values♥ SD = (|win – lose|)[P(win)P(lose)]
♣P(win) is the number of winning tickets divided by the total number of tickets
♣P(lose) = 1 - P(win)
♠ What is the SD of the box ?-$1 4 $4 1
$2
Standard Error
♠ The standard error gives a sense of how large the typical chance error (distance from the expected value) should be♥ In games of chance, the SE indicates
how “tight” a game is♣In games with a low SE, you are likely to
make near the expected value♣In games with a high SE, there is a chance
of making significantly more (or less) than the expected value
Standard Error
♠ Flip a fair coin and count the number of heads♥ What box models this game?
♥ How far off the expected number of heads should you expect to be in…
♣10 flips?♣100 flips?
0 1
1.58
5
Standard Error♠ Pay $1 to roll a fair die
♥ You win $5 if you roll an ace (1)♥ You lose the $1 otherwise
♠ What box models this game?
♠ How far off your expected gain should you expect to be in…♥ 1 game?♥ 5 games?
$2.24
$5.01
-$1 5 $5 1
The Law of Averages
♠ When playing a game repeatedly, as n increases, so do EVn and SEn
♥ However, SEn increases at a slower rate than EVn
♠ Consider the proportional expected value and standard error by dividing EVn and SEn by n♥ The proportional EV = EV1 regardless of n♥ The proportional SE decreases towards 0 as
n increases
The Law of Averages
♠ Flip a fair coin over and over and over and count the heads
n EVn SEn SEn/n
10 5 1.58 15.8%
100 50 5 5%
1000 500 15.8 1.6%
10000 5000 50 0.5%
The Law of Averages
♠ The tendency of the proportional SE towards 0 is an expression of the Law of Averages♥ In the long run, what should happen
does happen♥ Proportionally speaking, as the number
of plays increases it becomes less likely to be far from the expected value
The Central Limit Theorem♠ If you flip a fair coin once, the distribution for
the number of heads is♥ 1 with probability 1/2♥ 0 with probability 1/2
♠ This can be visualized with a probability histogram
n = 1
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
0 1
Number of Heads
Prob
abili
ty
The Central Limit Theorem
♠ As n increases, what happens to the histogram?♥ This illustrates the
Central Limit Theorem
n = 2
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
0 1 2
Number of Heads
Prob
abili
tyn = 10
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
0 1 2 3 4 5 6 7 8 9 10
Number of Heads
Prob
abili
ty
n = 100
0.00000%
2.00000%
4.00000%
6.00000%
8.00000%
10.00000%
0 10 20 30 40 50 60 70 80 90 10
Number of Heads
Prob
abili
ty
The Central Limit Theorem♠ The Central Limit Theorem (CLT) states
that if…♥ We play a game repeatedly♥ The individual plays are independent♥ The probability of winning is the same for
each play♠ Then if we play enough, the distribution
for the total number of times we win is approximately normal♥ Curve is centered on EVn♥ Spread measure is SEn
♠ Also holds if we are counting money won
The Central Limit Theorem
♠ The initial game can be as unbalanced as we like♥ Flip a weighted coin
♣ Probability of getting heads is 1/10♥ Win $8 if you flip heads♥ Lose $1 otherwise
n = 1
0.0%
20.0%
40.0%
60.0%
80.0%
100.0%
-1 8
Net Gain ($)
Prob
abili
ty
The Central Limit Theorem
♠ After enough plays, the gain is approximately normally distributed
n = 5
0.0%10.0%20.0%30.0%40.0%50.0%60.0%70.0%
-5 4 13 22 31 40
Net Gain ($)
Prob
abili
tyn = 25
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
-25 2 29 56 83 11
013
716
419
1
Net Gain ($)
Prob
abili
ty
n = 100
0.0%2.0%4.0%6.0%8.0%
10.0%12.0%14.0%
-100 -82
-64
-46
-28
-10 8 26 44 62 80 98
Net Gain ($)
Prob
abili
ty
The Central Limit Theorem
♠ The previous game was subfair♥ Had a negative expected value♥ Play a subfair game for too long and you are
very likely to lose money♠ A casino doesn’t care whether one person
plays a subfair game 1,000 times or 1,000 people play the game once♥ The casino still has a very high probability of
making money
The Central Limit Theorem
♠ Flip a weighted coin♥ Probability of getting heads is 1/10♥ Win $8 if you flip heads♥ Lose $1 otherwise
♠ What is the probability that you come out ahead in 25 plays?
♠ What is the probability that you come out ahead in 100 plays?
42.65%
35.56%
Roulette
♠ In roulette, the croupier spins a wheel with 38 colored and numbered slots and drops a ball onto the wheel♥ Players make bets on where the ball
will land, in terms of color or number♥ Each slot is the same width, so the ball
is equally likely to land in any given slot with probability 1/38 2.63%
Roulette♠ Players place their bets on the
corresponding position on the table
Red/Black1 to 1
Even/Odd1 to 1
1-18/19-361 to 1
Split17 to 1
Single Number35 to 1
Row11 to 1
Four Numbers8 to 1
2 Rows5 to 1
Column2 to 1
Section2 to 1
$
$$$$
$$
$$
$
Roulette
♠ One common bet is to place $1 on red♥ Pays 1 to 1
♣ If the ball falls in a red slot, you win $1♣ Otherwise, you lose your $1 bet
♥ There are 38 slots on the wheel♣ 18 are red♣ 18 are black♣ 2 are green
♠ What are the expected value and standard error for a single bet on red?
-$0.05 ± $1.00
Roulette
♠ One way of describing expected value is in terms of the house edge♥ In a 1 to 1 game, the house edge is
P(win) – P(lose)♣For roulette, the house edge is 5.26%
♠ Smart gamblers prefer games with a low house edge
Roulette
♠ Playing more is likely to cause you to lose even more money♥ This illustrates the Law of Averages
n EVn SEn
1 -$.05 $1.00
10 -$.53 $3.16
100 -$5.26 $9.99
1000 -$52.63 $31.58
10000 -$526.32 $99.86
Roulette
♠ Another betting option is to bet $1 on a single number♥ Pays 35 to 1
♣If the ball falls in the slot with your number, you win $35
♣Otherwise, you lose your $1 bet♥ There are 38 slots on the wheel
♠ What are the expected value and standard error for one single number bet?
-$0.05 ± $5.76
Roulette
♠ The single number bet is more volatile than the red bet♥ It takes more plays for the Law of Averages
to securely manifest a profit for the house
n EVn SEn1 -$.05 $1.0010 -$.53 $18.22
100 -$5.26 $57.631000 -$52.63 $182.23
10000 -$526.32 $576.26
Roulette
♠ If you bet $1 on red for 25 straight times, what is the probability that you come out (at least) even?
♠ If you bet $1 on single #17 for 25 straight times, what is the probability that you come out (at least) even?
40%
48%
Craps
♠ In craps, the action revolves around the repeated rolling of two dice by the shooter♥ Two stages to each round
♣ Come-out Roll♦ Shooter wins on 7 or 11♦ Shooter loses on 2, 3, or 12 (craps)
♣ Rest of round♦ If a 4, 5, 6, 8, 9, or 10 is rolled, that number is the
point♦ Shooter keeps rolling until the point is re-rolled
(shooter wins) or he/she rolls a 7 (shooter loses)
Craps♠ Players place their bets on the
corresponding position on the table♥ Common bets include
Pass1 to 1
Don’t Pass1 to 1
Come1 to 1
Don’t Come1 to 1
Craps
♠ Pass/Come, Don’t Pass/Don’t Come are some of the best bets in a casino in terms of house edge
♠ In the pass bet, the player places a bet on the pass line before the come out roll♥ If the shooter wins, so does the player
Craps: Pass Bet
♠ The probability of winning on a pass bet is equal to the probability that the shooter wins♥ Shooter wins if
♣Come out roll is a 7 or 11♣Shooter makes the point before a 7
♥ What is the probability of rolling a 7 or 11 on the come out roll?
8/36 ≈ 22.22%
Craps: Pass Bet
♠ The probability of making the point before a 7 depends on the point♥ If the point is 4, then the probability of
making a 4 before a seven is equal to the probability of rolling a 4 divided by the probability of rolling a 4 or a 7
♣This is because the other numbers don’t matter once the point is made
3/9 ≈ 33.33%
Craps: Pass Bet
♠ What is the probability of making the point when the point is…♥ 5?♥ 6?♥ 8?♥ 9?♥ 10?
♠ Note the symmetry
4/10 = 40%
3/9 ≈ 33.33%
4/10 = 40%
5/11 ≈ 45.45%
5/11 ≈ 45.45%
Craps: Pass Bet
♠ The probability of making a given point is conditional on establishing that point on the come out roll♥ Multiply the probability of making a
point by the probability of initially establishing it
♣This gives the probability of winning on a pass bet from a specific point
Craps: Pass Bet
♠ Then the probability of winning on a pass bet is…
♠ So the probability of losing on a pass bet is…
♠ This means the house edge is
8/36 + [(3/36)(3/9) + (4/36)(4/10) + (5/36)(5/11)](2) ≈ 49.29%
100% - 49.29% = 50.71%
49.29% - 50.71% = -1.42%
Craps: Don’t Pass Bet
♠ The don’t pass bet is similar to the pass bet♥ The player bets that the shooter will
lose♥ The bet pays 1 to 1 except when a 12
is rolled on the come out roll♣If 12 is rolled, the player and house tie
(bar)
Craps: Don’t Pass Bet
♠ The probability of winning on a don’t pass bet is equal to the probability that the shooter loses, minus half the probability of rolling a 12♥ Why half?
♠ Then the house edge for a don’t pass bet is
50.71% - (2.78%)/2 = 49.32%
50.68% - 49.32% = 1.36%
Craps: Don’t Pass Bet
♠ Note that a don’t pass bet is slightly better than a pass bet♥ House edge for pass bet is 1.42%♥ House edge for don’t pass bet is 1.36%
♠ However, most players will bet on pass in support of the shooter
Craps: Come Bets
♠ The come bet works exactly like the pass bet, except a player may place a come bet before any roll♥ The subsequent roll is treated as the
“come out” roll for that bet♠ The don’t come bet is similar to the
don’t pass bet, using the subsequent roll as the “come out” roll
Craps: Odds
♠ After a point is established, players may place additional bets called odds on their original bets ♥ Odds reduce the house edge even closer to 0♥ Most casinos offer odds, but at a limit
♣ 2x odds, 3x odds, etc…♥ If the odds are for pass/come, we say the
player takes odds♥ If the odds are for don’t pass/don’t come, we
say the player lays odds
Craps: Odds
♠ Odds are supplements to the original bet♥ The payoff for an odds bet depends on
the established point♥ For each point, the payoff is set so that
the house edge on the odds bet is 0%
Craps: Odds♠ If the point is a 4 (or 10), then the
probability that the shooter wins is 3/9 ≈ 33.33%♥ The payoff for taking odds on 4 (or 10) is then 2 to 1
♠ If the point is a 5 (or 9), then the probability that the shooter wins is 4/10 = 40%♥ The payoff for taking odds on 5 (or 9) is then 3 to 2
♠ If the point is a 6 (or 8), then the probability that the shooter wins is 5/11 ≈ 45.45%♥ The payoff for taking odds on 6 (or 8) is then 6 to 5
Craps: Odds
♠ Similarly, the payoffs for laying odds are reversed, since a player laying odds is betting on a 7 coming first♥ The payoff for laying odds on 4 (or 10)
is then 1 to 2♥ The payoff for laying odds on 5 (or 9) is
then 2 to 3♥ The payoff for laying odds on 6 (or 8) is
then 5 to 6
Craps: Odds
♠ Keep in mind that although odds bets are fair-value bets, you must make a negative expectation bet in order to play them♥ The house still has an edge due to the
initial bet, but the odds bet dilutes the edge
Craps: Odds
♠ Suppose you place $2 on pass at a table with 2x odds♥ Come out roll establishes a point of 5♥ You take $4 odds on your pass
♠ Shooter eventually rolls a 5♥ You win $2 for your original bet and $6
for the odds bet
Craps
♠ Suppose a player bets $1 on pass for 25 straight rounds♥ What is the probability that she comes
out (at least) even?
47%
Summary
♠ Many chance processes can be modeled by drawing from a box filled with marked tickets♥ The value on the ticket represents the value
of the outcome♠ The expected value of an outcome is the
weighted average of the tickets in the box♥ Gives a prediction for the outcome of the
game♥ A game where EV = 0 is said to be fair
Summary
♠ The standard error gives a sense of how far off the expected value we might expect to be♥ The smaller the SE, the more likely we
will be close to the EV♠ Both the EV and SE depend on the
number of times we play
Summary
♠ As the number of plays increases, the probability of being proportionally close to the expected value also increases♥ This is the Law of Averages
♠ If we play enough times, the random variable representing our net winnings is approximately normal♥ True regardless of the initial probability of
winning
Summary
♠ Roulette and craps are two popular chance games in casinos♥ Both games have a negative expected
value, or house edge♥ Intelligent bets are those with small
house edges or high SE’s